### The mathematical approach

**Even when there is no drift, so that μ = 0, the expected value of the lognormal distribution is affected by the volatility.**

Thus, to simulate paths with truly zero drift, we have to subtract .5*σ² from the drift.

So now we have learned something many people overlook or forget: when you are simulating lognormal returns, you always have to subtract away half the variance from the drift. This is far and away the most common mistake in simulating returns.

### The empirical approach

Just take those 1,000 paths we simulated and compute the average

Also see:

*n (Moontower)***Lessons From The .50 Delta Optio**